Questions tagged [triangulated-categories]

A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.

Filter by
Sorted by
Tagged with
1
vote
1answer
86 views

Computing Ext groups in a functor stable $\infty$-category

Let $I$ be a small category and $\mathcal{D}=D^b_\infty(\mathbb{Z})$ the bounded derived $\infty$-category of abelian groups. Consider the $\infty$-category $\mathcal{C}:=\mathrm{Fun}(I,\mathcal{D})$. ...
1
vote
0answers
83 views

Can morphisms of Mayer-Vietoris triangles be completed into a $3\times 3$ square?

Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ and $\mathcal{F}'$ ...
2
votes
0answers
65 views

Cone of a morphism of complexes that are concentrated in degree $0$ and $1$

Let $R$ be a ring and $f:A\to A'$ and $g:B\to B'$ be morphisms of $R$-modules. Let $h:C_{\bullet}\to C_{\bullet}'$ be a morphism of $R$-module complexes fitting in a morphism of distinguished ...
3
votes
1answer
103 views

Smallness condition for augmented algebras

I'm not sure this question is research level question. Sorry in advance. Hypothesis $k$ is a commutative ring. $A$ is an augmented $k$-algebra. $A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...
3
votes
0answers
134 views

Who introduced the heart ($\mathcal{C}^\heartsuit$) notation in the context of $t$-structures on triangulated categories?

In the context of $t$-structures ([Wikipedia], [nLab], [Notes I], [Notes II], [HA, Definition 1.2.1.11)], [BBD, Définition 1.3.1]), one often writes $\mathcal{C}^\heartsuit$ for the heart of a ...
1
vote
1answer
198 views

Homotopy equivalences and Mapping Cones

Edit: Version 2: Suppose that $A,B,C$ are chain complexes and $f: A \rightarrow B$ is a chain map. Suppose that there is a homotopy equivalence $$ \text{Cone}(f: A \rightarrow B) \simeq C.$$ The chain ...
6
votes
1answer
394 views

“Universal” triangulated category

Let $\mathcal{C}$ be some category. One way to map this category into a triangulated category is to take the category of simplicial objects $s\mathcal{C}$ (which is an $\infty$-category), take its ...
5
votes
0answers
237 views

Proof in the realm of $\infty$ categories

I have recently started learning the language of $\infty$-categories. My approach is more to their use, rather than for their own sake. For this reason, as I feel I reached a good understanding of the ...
4
votes
1answer
144 views

Does localization at quasi-isomorphisms imply homotopy invariance?

Usually, the derived category of some abelian category $A$ (I'm happy already with $A$-mod) is defined first taking chain complexes up to homotopy, and then localize at quasi-isomorphisms. My question ...
2
votes
1answer
99 views

Admissibility of intersection of subcategories

Let $\mathscr{T}$ be a triangulated category, and $\mathscr{A}$ be a right admissible subcategory, which means that $i_{\mathscr{A}} : \mathscr{A} \rightarrow \mathscr{T}$ has a right adjoint $i_{\...
1
vote
1answer
447 views

idea and intuition behind triangulated category [closed]

I have some trouble in understanding the significance of some axiom of triangulated category. if someone could explain me each axiom with some intuition,and explain me the intuition behind the ...
1
vote
1answer
294 views

Why are Serre functors always exact?

Let $k$ be a field and $\mathcal{T}$ be a $k$-linear triangulated category with finite dimensional spaces of morphisms. Bondal and Kapranov proved that every Serre functor on $\mathcal{T}$ is exact (...
9
votes
2answers
274 views

When is the homotopy category of an accessible $\infty$-category accessible?

Let $\mathcal C$ be an accessible $\infty$-category, and let $ho(\mathcal C)$ be its homotopy category. I can think of two "trivial" reasons for $ho(\mathcal C)$ to be accessible: $ho(\mathcal C) = \...
16
votes
0answers
514 views

The Octahedral Axiom in group theory

$\require{AMScd}$Here are two results about groups: (The third isomorphism theorem) Suppose that I have $A \triangleleft B \triangleleft C$ and $A \triangleleft C$. Then $C/B \cong (C/A)/(B/A)$. ...
4
votes
0answers
96 views

Morphism in a Verdier quotient

Let $\mathcal{T}$ be a triangulated category and take $\mathcal{S}$ a triangulated subcategory. Consider the Verdier quotient $\mathcal{T} \left/ \mathcal{S} \right.$, morphisms in this category are ...
3
votes
1answer
213 views

A question on the proof of $D^b(coh(X))\simeq D^b_{coh}(Qcoh(X))$

Proposition 3.5 of "Fourier-Mukai Transforms in Algebraic Geometry" by Huybrechts claims that the is an equivalence of categories $$ D^b(coh(X))\overset{\sim}{\to} D^b_{coh}(Qcoh(X)) $$ where $D^b(coh(...
3
votes
0answers
152 views

Do we have $D^b_{coh}(X)\simeq D^b(coh(X))$ for a compact complex manifold $X$?

Let $X$ be a compact complex manifold and $\mathcal{O}_X$ be the structure sheaf of holomorphic functions. We call a sheaf of $\mathcal{O}_X$-module $\mathcal{F}$ coherent if it satisfies the ...
5
votes
1answer
262 views

Can homotopy colimits recover cohomology sheaves?

The question is basically the one outlined in the title. Let $\mathcal{T}$ be a triangulated category containing infinite direct sums (e.g. $D_{qc}(X)$ for some separated, finite type over a field $k$,...
2
votes
1answer
101 views

When is the heart of a triangulated category Grothendieck?

Are there conditions which guarantee that the heart of a triangulated category is Grothendieck? Is the compatibility between the t-structure with filtered colimits enough?
12
votes
1answer
598 views

A concrete example of the deficiency of triangulated categories?

There seems to be a general sentiment that triangulated categories are not the "correct" notion to use because mapping cones of morphisms are unique, but only up to non-unique isomorphism. Does ...
1
vote
0answers
67 views

Equivalences between $D^b(\mathcal{B})$ and $\mathcal{D}_{\mathcal{B}}$, the triangulated category generated by $\mathcal{B}$

In the paper "Finite dimensional algebras and highest weight categories" of Cline, Parshall and Scott is stated as follows: Let $\mathcal{B}$ be an abelian subcategory of a triangulated category $\...
23
votes
3answers
2k views

Replacing triangulated categories with something better

Gelfand and Manin in their 1988 book on homological algebra write that the non-functoriality of cones means that "something is going wrong in the axioms of a triangulated category. Unfortunately at ...
6
votes
0answers
203 views

Example of a tensor triangulated category with two different monoidal t-structures?

What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures? While I'm at it: is there an example of a tensor ...
5
votes
1answer
218 views

Computing a cone in a $\otimes$-triangulated category

I have a $\otimes$-triangulated category $\mathcal T$ and two triangles in $\mathcal T$: $$ x_0\to x_1\to c_x\to \Sigma x_0\ \ \ \text{and}\ \ \ y_0\to y_1\to c_y\to \Sigma y_0. $$ Consider the ...
9
votes
0answers
332 views

Kontsevich's derived noncommutative geometry and Rosenberg's noncommutative 'spaces'

It appears to me (though I may be wrong) that the common opinion is that the main difference between two is that Rosenberg's version of noncommutative algebraic geometry mainly concerns as ...
4
votes
2answers
243 views

Are hearts of all $t$-structures on smashing triangulated categories closed with respect to coproducts (also)?

Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with ...
6
votes
1answer
379 views

Homotopy pullbacks and pushouts in stable model categories

There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...
1
vote
0answers
110 views

Which set of compact objects generates the subcategory of a compactly generated stable model category?

I couldn't find any info on what set of compact objects generates the following subcategory: Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group ...
5
votes
1answer
331 views

Are there universal homological functors?

There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable? That is, for each small abelian ...
6
votes
0answers
141 views

Abelianization derivator

About ten-fifteen years ago, when the theory of abstract triangulated categories reached a culminating point (after the publication of Neeman's book http://hopf.math.purdue.edu/Neeman/triangulatedcats....
4
votes
0answers
156 views

Serre functors for non-proper categories

One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...
3
votes
1answer
200 views

Filtered triangulated category examples

I am reading Beilison Ginsburg Schechtman's "Koszul duality". In the section 1.3, they introduced the notion filtered triangulated categories with only one example, considering an abelian category ...
3
votes
0answers
137 views

Weak generators of the right-bounded derived category of a finite-dimensional algebra

The setup: Let $A$ be a finite-dimensional $k$-algebra over some field $k$. Let $\mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...
1
vote
0answers
59 views

Filtrations of spectra related to cellular ones and singular homology

I would like to study filtrations of spectra (i.e., objects of the "topological" stable homotopy category $SH$; a filtration of a spectrum $E$ is a sequence of compatible maps $E_{\le i}\to E$) whose ...
4
votes
2answers
312 views

When is $\Omega^1$ an equivalence?

Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules. When is the functor $\Omega^1 : \underline{...
12
votes
2answers
426 views

Is there any significance to Bousfield localization in the non-derived context?

The term "Bousfield localization" of a category $C$ is used in roughly two different ways: There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...
1
vote
0answers
129 views

Highest weight category and weight structures

In various branches of representation theory, there is a notion of highest weight category. On the other hand there is a notion of weight structure on a triangulated category $C$ (introduced by ...
4
votes
0answers
152 views

Generators of unbounded derived categories of (quasi-)coherent sheaves

An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...
7
votes
0answers
118 views

Why do we need the negative sign in TR2 (the “turning triangle” axiom) in the definition of triangulated categories?

In the TR2 of the definition of triangulated categories, we add a negative sign to an arrow when we turn the triangles. What is the significance/motivation of that negative sign ($-u[1]$ as in the ...
10
votes
1answer
323 views

Vanishing natural transformation exact triangle

This question is a follow-up to this question I asked some time ago. Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{C}$. Let $\omega \in H^{n}(X,K_X)$, $\omega \neq 0$. Let $$A ...
8
votes
1answer
641 views

Is the derived category of $A$-dg-modules as a dg-category coincide with the ordinary definition of derived category?

Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We ...
8
votes
2answers
498 views

If the homotopy category is well-generated, must the $\infty$-category be presentable?

Suppose $\mathcal{C}$ is a stable $\infty$-category whose homotopy category is a well-generated triangulated category in the sense of Neeman's book. Must $\mathcal{C}$ be a presentable $\infty$-...
6
votes
1answer
580 views

A question on Voevodsky´s categories

I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions: 1.- ...
2
votes
0answers
166 views

(Middling) good morphisms of triangles

Neeman in his article "Some new axioms for triangulated categories" calls a morphism of distinguished triangles $$\require{AMScd} \begin{CD} X @>>> Y @>>> Z @>>> X [1] \\ @...
1
vote
1answer
161 views

Comparing self-equivalences of a triangulated category and automorphisms of its Grothendieck group

There is a homomorphism from the group of (isomorphism classes of) self-equivalences of a triangulated category to the automorphism group of its Grothendieck group. Is this homomorphism surjective? If ...
5
votes
0answers
209 views

Prime spectrum of the derived category of holonomic $\mathcal{D}$-modules?

Let $X$ be a smooth algebraic (/projective if it simplifies things considerably) variety over $\mathbb{C}$ and consider the derived category $\mathcal{C}=D_h^b(\mathcal{D}_X)$ of bounded complexes of $...
1
vote
0answers
68 views

Triangulated categories generated by a collection of submodules

In the book "Cohomology Rings of Finite Groups", by Carlson- Townsley - Elizondo there is the following corollary The category $\mathsf{stmod}_{\mathbb{k}G}$ is generated as a triangulated category ...
1
vote
1answer
146 views

Generating $K^b(\mathrm{proj})$ as a triangulated category from a full subcategory

Let $K^b(\mathrm{proj}\, A)$ be the bounded homotopy category of chain complexes over $\mathrm{proj}\, A$. In Rickard's paper 'Derived categories and stable equivalence', he defines a tilting complex ...
1
vote
0answers
43 views

Extending a natural transformation using a distinguished triangle

$\require{AMScd}$ Let $\mathcal{T}$ be a triangulated category, and $\mathcal{S}$ a full subcategory of $\mathcal{T}$ (which is not triangulated). Let $F, G: \mathcal{T} \to \mathcal{T}$ be two ...
1
vote
0answers
65 views

Techniques for Showing Triviality of K_1 of a Higher Category

Suppose $\cal{C}$ is a small stable $\infty$-category. Then, we have its K-theory spectrum $K(\cal{C})$ that gives us K-theory groups $K_n(\cal{C})$ by taking stable homotopy groups. There are ...

1
2 3 4 5